From a point perpendicular tangents are drawn on the ellipse $x^2+2y^2=2$. The chord of contact touches a circle concentric with ellipse. Find ratio of min and max area of circle
Let the point from which tangents are drawn be $(h,k)$
Then the locus of that point will be $$h^2+k^2=3$$
Also the chord of contact is $$\frac{hx}{2}+ky-(\frac {h^2}{2}+k^2)=0$$
Let the circle be $$x^2+y^2=a^2$$
Then the tangent to this circle is $$y=\frac{-h}{2k}x\pm a\sqrt{1+\frac{h^2}{4k^2}}$$
$$hx+2ky \mp a\sqrt{4k^2+h^2}=0$$ Now I could equate the $c$ term of the linear equations, but that’s a very lengthy process, so I am convinced I am approaching the question wrong. How should I do it right?